Modular Arithmetic Lesson 4.6

This section, we are grouping numbers, based on their reminders. These are said to be in the same congruence class. 18 ≡ 22 (mod 4)  bc their remainders are the same

How many congruence classes are there in modulo 7? There are 7 congruence classes! R6 R0

Congruence Theorem For all integers a and b and all positive integers m, a ≡ b (mod m) iff m is a factor of a – b. ISBN and credit card check numbers are determined in some way by modular arithmetic.

Example 1 The final digit of a 12-digit Universal Product Code (UPC) is a check digit. Suppose the digits of a UPS are represented by X1X2X3X4X5X6-X7X8X9X10X11Xc To calculate the check digit Xc, first compute 3(X1+X3+X5+X7+X9+X11) + (X2+X4+X6+X8+X10) (mod 10). If this number is zero, then Xc=0, If not, subtract the number from 10. Determine the check digit of 300219 – 09529 3(3 + 0 + 1 + 0 + 5 + 9) + (0 + 2 + 9 + 9 + 2) = 3(18) + (22) = 76 ÷ 10 = 7 r. 6 So Xc=4

Example 2 Use modular arithmetic to explain why a date that falls on Friday this year will fall on Wednesday four years from now. 365 days in a year + 1 leap year! 1461 days ÷ 7 days in a week = 208 weeks r. 5 Friday – r. 0 Tuesday – r. 4 Sat. – r.1 Wednesday – r. 5 Sun. – r.2 Thursday – r. 6 Monday – r.3

Example 3 Find the smallest positive value of n for which n – 32 ≡ 75 (mod 11) n ≡ 107 (mod 11) Congruence class of R8 n = 8

Homework Pages 261 – 262 1-3, 5-11, 19